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Some hex-splines living on the hexagonal lattice. Click to magnify
Hex-splines: A novel spline family for hexagonal lattices

The standard approach to represent two-dimensional data uses orthogonal lattices. Nevertheless, hexagonal lattices provide several advantages, including a higher degree of symmetry and a better packing density.

Hex-splines are a new type of bivariate splines especially designed for hexagonal lattices. Inspired by the indicator function of the Voronoi cell, they are able to preserve the isotropy of the hexagonal lattice (as opposed to their B-spline counterparts). They can be constructed for any order and are piecewise polynomial (on a triangular mesh). Analytical formulas have been worked out in both spatial and Fourier domains. For orthogonal lattices, the hex-splines revert to the classical tensor-product B-splines.

Similar to B-splines, hex-splines can be useful for various areas of engineering. For example in image processing they are applied to obtain better resampling algorithms for printing (e.g., gravure printing and CMYK color printing) and machine vision (edge detection and pattern recognition). Some researchers have studied the design of advanced smart CMOS sensors using the hexagonal arrangement. The hex-splines could be used to design better operators for data obtained by such sensory. A recent and noteworthy example is the 4th generation superCCD of FujiFilm that uses pixel elements in a hexagonal-like arrangement. Other applications might include the modeling and mimicking of the human vision system and the dynamic simulation of micromagnetic fields. We refer to our paper for a list of relevant references.

Example: the eye of "Lena"

Magnification using first order

Magnification using second order

Magnification using third order

First horizontal derivative for second order
Dimitri Van De Ville
Maple worksheets to obtain the analytical formula of any hex-spline (any order, regular, non-regular, derivatives, and so on)
Important note:
You are free to use this software for research purposes, but you should not redistribute it without our consent. In addition, we expect you to include adequate citations and acknowledgments whenever you present or publish results that are based on it.

Dimitri Van De Ville, Thierry Blu, Michael Unser, Wilfried Philips, Ignace Lemahieu, Rik Van de Walle, "Hex-splines: A novel spline family for hexagonal lattices," IEEE Transactions on Image Processing, 13(6), pp. 758-772, June 2004.

Related articles

First-order hex-spline

Second-order hex-spline

Third-order hex-spline